Information-Theoretic Lyapunov Function V

• The topic defines information-theoretic Lyapunov functions as nonnegative, convex functionals that measure the divergence between a system's state and its equilibrium using entropy-related metrics.
• They enable the analysis of exponential stability in systems such as nonlinear PDEs, consensus dynamics, and non-reversible stochastic processes through dissipation estimates and spectral gap techniques.
• Applications include coupled diffusion systems and gradient flows, unifying methods from large deviations, statistical mechanics, and entropy production for rigorous stability analysis.

An information-theoretic Lyapunov function $V$ is a nonnegative, typically convex functional designed to quantify the divergence or "distance" between the current state of a system and its equilibrium, with the key property that $V$ decreases (or is non-increasing) along the system's evolution. Such functionals—rooted in the entropy, Kullback–Leibler divergence, Fisher information, and large deviation principles—have become central to the analysis of relaxation and stability in nonlinear partial differential equations, consensus dynamics, hydrodynamic limits of interacting particle systems, and non-reversible stochastic flows. Unlike classical quadratic Lyapunov functions, they are well-suited to non-Euclidean geometries, open systems, and non-equilibrium stationary measures, and they enable sharp, quantifiable rates of exponential convergence under minimal regularity assumptions (Bodineau et al., 2013, Feng et al., 2020, Mangesius et al., 2014).

1. General Constructions of Information-Theoretic Lyapunov Functionals

The construction begins with a reference stationary profile (or invariant distribution) $f_\infty$ (respectively, $\pi$), and a strictly increasing, convex functional depending on the system's configuration $f$ (density/profile); see (Bodineau et al., 2013). Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $\nu$ a reference measure (often the Lebesgue measure), $A(x)$ a symmetric positive-definite diffusion matrix, $E(x)$ a drift, and $\sigma$ a strictly increasing $C^2$ nonlinearity. The core evolution equation is: $$\partial_t f = -\nabla\cdot[A(x)\nabla(\sigma(f)\nu) + E(x)\sigma(f)\nu]/\nu,$$ with Dirichlet boundary data $f|_{\partial\Omega}=f_b(x)>0$, and stationary $f_\infty$ solving: $$-\nabla\cdot[A\nabla(\sigma(f_\infty)\nu) + E\sigma(f_\infty)\nu] = 0, \quad f_\infty|_{\partial\Omega}=f_b.$$

Two broad families of Lyapunov functionals emerge, parameterized by strictly convex, $C^2$ generating functions $\Phi, \Psi$: $$H_\Phi(f \| f_\infty) = \int_\Omega \int_{s=f_\infty(x)}^{f(x)} \Phi'(\sigma(s)/\sigma(f_\infty(x))) \,s\, d\nu(x),$$

$$N_\Psi(f \| f_\infty) = \int_\Omega \int_{s=f_\infty(x)}^{f(x)} \Psi'(\sigma(s)-\sigma(f_\infty(x)))\,s\, d\nu(x),$$

with specific choices recovering classical quantities: e.g., for $\sigma(s)=s$, $\Phi(z) = z \log z - z + 1$, $H(f \| f_\infty)$ yields the relative entropy (Kullback–Leibler divergence); for power-law $\sigma(s)=s^m$, $\Psi(z) = z^2/2$, $N(f \| f_\infty)$ reduces to a quadratic generalized entropy (Bodineau et al., 2013).

2. Large Deviations, Gradient Flows, and Statistical Mechanics Connections

The information-theoretic Lyapunov structure generalizes the notion of large-deviation rate functionals from stochastic interacting particle systems to continuum nonlinear PDE. In particular, for the zero-range process, the large-deviation functional for empirical densities is precisely of the $H_\Phi$ form; for the Ginzburg–Landau dynamics, the $N_\Psi$ form is canonical. These functionals also serve as potentials for gradient flows in generalized (non-Euclidean) geometries: additive convex Lyapunov functions (including Kullback–Leibler divergence) can act as Riemannian potentials for consensus and Markov systems, as shown in (Mangesius et al., 2014).

For consensus systems $\dot x = -Lx$ with $L$ irreducible, any strictly convex function $H$ gives rise to the Lyapunov candidate $V(x) = \sum_i q_i H(\rho_i)$ (where $q$ is the left Perron vector and $\rho_i$ are normalized densities), with strict Lyapunov property and a Riemannian gradient flow representation: $\dot x = -G^{-1}(x) \nabla V(x),$ with $G^{-1}(x)$ a state-dependent metric operator constructed from $H'$, generalizing the logarithmic mean in the classical entropy case (Mangesius et al., 2014).

3. Dissipation Estimates and Entropy–Entropy-Production Inequalities

A central analytic tool is the relation between a Lyapunov functional and its dissipation rate. For the porous medium equation ($\sigma(s) = s^m$, $A = I$, $\nu = dx$), the functional

$$V[f] = \int_\Omega \left[ \frac{f^{m+1} - f_\infty^{m+1}}{m+1} - (f - f_\infty) f_\infty^m \right] dx$$

has dissipation

$$D[f] = -\frac{d}{dt}V[f] = \int_\Omega |\nabla (f^m - f_\infty^m)|^2 dx,$$

and using the Dirichlet Poincaré inequality, one establishes the linear bound $D[f] \geq \alpha V[f]$, $\alpha$ determined by the first Dirichlet eigenvalue and bounds on $f_\infty$. Thus, $V[f]$ decays exponentially: $V[f(t)] \leq V[f(0)] e^{-\alpha t}$ (Bodineau et al., 2013).

Similarly, in non-reversible stochastic differential equations driven by

$dX_t = b(X_t)\,dt + \sqrt{2}\,dB_t,$

the Fisher information

$$V(\mu) = I(\mu \| \pi) = \int_\Omega \|\nabla \log (p/\pi)\|^2\, p\, dx$$

serves as a Lyapunov functional. Under a suitable spectral gap on a curvature-type tensor $\mathfrak{R}(x)$ (incorporating both the stationary measure $\pi$ and non-gradient components of the drift $b$), $I(p_t \| \pi)$ decays exponentially, and, by the de Bruijn identity and standard information-theoretic inequalities, exponential convergence propagates in Kullback–Leibler, Wasserstein, and $L^1$ norms (Feng et al., 2020).

4. Extensions: Coupled Systems, Non-Reversible Processes, and Gradient Flows

Information-theoretic Lyapunov functionals extend naturally to coupled nonlinear diffusion systems, under compatibility conditions for product-form invariants. For systems of the form

$$\partial_t f^k = -\nabla\cdot\left[A_k \nabla \left(\sigma^k(f^1, f^2)\nu\right)\right], \quad k=1,2,$$

an additive or quadratic Lyapunov structure persists, provided the nonlinearity cross-derivatives satisfy compatibility (e.g., $\partial_2 \ln \sigma^1 = \partial_1 \ln \sigma^2$), ensuring dissipation and convergence toward coupled equilibria (Bodineau et al., 2013).

For jump Markov processes,

$$\partial_t \mu = \int [K(y, \cdot) \sigma(\mu/\nu)(y) - K(\cdot, y) \sigma(\mu/\nu)(\cdot)]\, \nu(dy),$$

Lyapunov property follows for the same $H_\Phi, N_\Psi$ forms, manifest in the non-positivity of integrals using convexity and the symmetrization afforded by reversibility or detailed balance (Bodineau et al., 2013).

In the gradient-flow context, convex additive Lyapunov functions unify energy viewpoints across domains: they serve as free energy in chemical kinetics, stored electric energy in circuit theory, and entropy or information divergence in probabilistic and consensus settings, all supporting generalized geometric structures for stability and convergence analyses (Mangesius et al., 2014).

5. Information-Theoretic Lyapunov Functions in Non-Reversible Stochastic Analysis

The recent development of information-theoretic Lyapunov methods in non-reversible, non-gradient stochastic dynamics is anchored on Gamma-calculus techniques and spectral estimates. For the non-reversible Fokker–Planck equation, the time derivative of Fisher information is computed as a negative quadratic form in gradients of the log density ratio, corrected by a curvature matrix incorporating both reversible and non-reversible contributions. The sufficient curvature lower bound yields explicit exponential decay. The approach generalizes log-Sobolev-based proofs and supplies explicit contraction rates in the absence of detailed balance (Feng et al., 2020).

By tensorizing with the entropy, this Lyapunov approach enables a full suite of convergence results: exponential rates in Kullback–Leibler divergence, Wasserstein distance, and $L^1$ norm, furnishing practical quantitative convergence controls for non-reversible MCMC and related stochastic numerical algorithms.

6. Synthesis and Unification Across Models

Information-theoretic Lyapunov functions encapsulate a systematic method bridging large deviation theory, entropy dissipation, stability of PDEs, and dynamical systems on discrete and continuous state spaces. They accommodate nonlinear, non-reversible, open, and interacting settings, where classical quadratic methods are insufficient. The theory provides both the Lyapunov structure and, crucially, quantitative exponential decay via entropy-production inequalities and spectral gap estimates, unifying disparate fields from nonequilibrium statistical physics, stochastic processes, diffusion equations, and networked consensus systems (Bodineau et al., 2013, Mangesius et al., 2014, Feng et al., 2020).

7. Tabular Summary of Key Functional Forms

Functional Name	Formulation	Typical Context
Relative Entropy $H$	$\int f \log(f/f_\infty) - f + f_\infty\,dx$	Linear drift-diffusion, Markov chains
Generalized Entropy $N$	$$\int \left[\frac{f^{m+1} - f_\infty^{m+1}}{m+1} - (f - f_\infty)f_\infty^m\right] dx$$	Porous medium equation
Fisher Information $I$	$\int \|\nabla \log(p/\pi)\|^2\, p\,dx$	Non-reversible SDEs
Additive Convex $V$	$\sum_i q_i H(\rho_i)$	Consensus, chemical reactions

These functionals are strictly Lyapunov under appropriate convexity and regularity assumptions, and their construction and analysis constitute the current state-of-the-art in entropy-method-based quantitative stability analysis of complex dynamical systems (Bodineau et al., 2013, Feng et al., 2020, Mangesius et al., 2014).